[Since this concerns gtr, followups should go to sci.physics.relativity]
On Mon, 23 Nov 1998 patanie@my-dejanews.com wrote:
> An interesting question to investigate,following the discovery of Miguel
> Alcubierre possible "travelling bubbles" of Space,at arbitrary speeds,is this
> one: what are the limits to the size of Alcubierre "bubbles" of Space?
An even more basic question is: can they exist at all? If you search the
Los Alamos preprint server you will find a dozen or so relevant preprints,
all tending to cast rather cold water on the existence of such "warps".
The problem which is easiest to understand involves energy: even a quite
modest "warp" turns out to require stupendous energies to create and
sustain itself.
Another problem is that the energy densities in a "warp" are typically
-negative- and therefore violate the energy conditions of gtr. Note that
Visser at al. have proven a fairly general "censorship theorem" forbidding
warps on the grounds that they must violate the energy conditions of gtr,
at least subject to some assumptions. See their preprint on the Los
Alamos server for details.
Negative energy is classically forbidden, but permitted (for sufficienty
short tiems) in QM. However, computations (not entirely rigorous)
employing such "quantum inequalties" again seem to indicate that
superluminal warps probably cannot exist because they would violate these
inequalities.
Some authors feel that the existence of horizons associated with
superluminal warps is another problem, although I think this viewpoint
might be erroneous.
I have observed in previous postings to sci.physics.relativity that using
Alcubierre's basic "bump function trick", it is easy to create warps in a
fluid or dust background rather than a vacuum background, and then one can
avoid violations of the energy conditions. However, the energy conditions
then impose new and even more stringent speed limits. Nonetheless, even
very slowly moving warp bubbles (or slowly rotating warp tubes) are
interesting, in my view, because they involve "inertialess transport" and
therefore seem relevant to Mach's principle, one of the most venerable
themes in gtr.
> If interstellar travel is possible through Alcubierre bubbles,then this would
> have important consequences for the present SETI research based on archaic
> radio signals. Advanced civilisations would have no necessity to use radio
> signals for interstellar communication but would use what I would call
> "Alcubierre interstellar posting" !
By my estimation, the (negative) energy E required (classically) to
sustain a tin-can shaped warp bubble moving in Minkowksi spacetime is on
the order of
-E ~ L T v^2/6
where v is the root mean square speed of the warp (it requires some
discussion to clarify what "speed of the warp" means, but never mind) and
T is the proper time (of the inhabitants of the bubble) of the duration of
the bubbles existence, and where L is the length of the warp bubble.
This estimate is fairly general; to derive it, start with the Einstein
tensor of my axially-symmetric warp bubble metric
ds^2 = dt^2 - [dz - V(t)(1 - F(z)G(r))dt]^2 - dr^2 - r^2 (d theta)^2
where F:R --> [0,1] and G:[0,infty) --> [0,1] are arbitrary bump functions
and where V(t) is an arbitrary smooth function giving the "speed" of the
warp bubble as a function of time, and approximate F and G by piecewise
linear functions in the integral of the energy T^{00} over the "world
tube" of the warp, and finally, make some simplifying assumptions; for
instance, assume that the warp walls are comparable in thickness to the
length and diameter of the bubble. Note that the resulting expression for
-E should not be much affected by changing the geometry of the bubble
(e.g. from a tin-can to a spherical bubble). Now, the trouble is that v
is squared in the expression LTv^2/6, and v is the one thing you can't
reduce if you want to travel -fast- :-)
> If Alcubierre bubbles of tiny sizes(such as the size of a hydrogen molecule
> to a big parcel size!) could exist,it might be possible to place information
> inside and post it immediately!
Forget people or hydrogen molecules: how about a single -electron-? To
transport an electron from here to Andromeda in about one year, take L ~
10^(-12) cm (slightly larger than the classical electron radius), with
warp walls about that thick, and take v ~ 10^6 (the speed required for a
journey to Andromeda taking on the order of one year) and take T ~ 10^18
cm (about 1 year). This gives
-E ~ 10^(-12) 10^18 (10^6)^2/6 ~ 10^17 cm
or on the order of ten times the mass of our entire galaxy. Ouch.
(See for instance Cuifolini & Wheeler, Gravitation and Inertia, or Wald,
General Relativity, for the "natural units" used in these computations.)
You might think that we can try sending Morse code using tiny warp
bubbles. However, even a Planck length sized warp bubble making the
journey from here to Andromeda in one year would require -E ~ 10^(-3) cm
or a few hundreths of the mass of the Earth. So if we sacrifice Venus we
could in principle send a very brief message to our nearest neighbors (I'm
assuming we could convert the mass of Venus into -negative- energy,
however that might work).
> I think Miguel Alcubierre should seriously think of such possibilities and
> investigate how such interstellar "letters" or "parcels" would look like.
He has been quoted as saying that the preprints I alluded to above have
convinced him that superluminal warp bubbles cannot be constructed.
I think a fairer assessment using the best theoretical evidence now
available is that superluminal warps probably cannot exist during the
present epoch of the universe, but just might have been plentiful during a
much earlier epoch. (But my only reason for saying that is that as far as
I can see, the classical theory of gtr doesn't forbid it, if the ambient
mass-energy density is high enough.)
> Another point: when such parcels would meet planets or stars on their
> way what would happen? Could they just go through undamaged?
As a matter of fact, warp bubbles of the type discussed here would not
make their presence known by the usual long-distance gravitational
effects, since the curvatures vanish outside the warp walls. However,
quantum phenomena might be very significant (but harder to compute
reliably).
> If yes then it should be possible to observe such occurences and
> detect them through their specific signature.
It is not hard to see that warp bubbles decellerating back to subluminal
speed should emit a flash of light in the direction of motion. However, I
haven't attempted to put an upper bound on the implied frequency of such
events, using the fact that they (apparently) haven't been observed. (So
many ideas, so little time to think about them...)
Chris Hillman
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Thanks!